Abstract : We consider the problem of computing exact sums of squares (SOS)
decompositions for certain classes of non-negative multivariate
polynomials, relying on semidefinite programming (SDP) solvers.
We provide a hybrid numeric-symbolic algorithm computing exact rational
SOS decompositions with rational coefficients for polynomials lying in the
interior of the SOS cone. The first step of this algorithm computes an
approximate SOS decomposition for a perturbation of the input polynomial with an
arbitrary-precision SDP solver. Next, an exact SOS decomposition is obtained thanks to
the perturbation terms and a compensation phenomenon. We prove that bit
complexity estimates on output size and runtime are both polynomial in the
degree of the input polynomial and singly exponential in the number of
variables. Next, we apply this algorithm to compute exact Reznick,
Hilbert-Artin's representation and Putinar's representations respectively for
positive definite forms and positive polynomials over basic compact
semi-algebraic sets. We also report on practical experiments done with the
implementation of these algorithms and existing alternatives such as the
critical point method and cylindrical algebraic decomposition.